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## WHAT IS IT?

Deffuant, Guillaume, Timoteo Carletti, and Sylvie Huet. “The Leviathan Model: Absolute Dominance, Generalised Distrust, Small Worlds and Other Patterns Emerging from Combining Vanity with Opinion Propagation.” Journal of Artificial Societies and Social Simulation 16, no. 1 (2012): 5. Available here: http://jasss.soc.surrey.ac.uk/16/1/5.html

Here is a copy of the abstract: "We propose an opinion dynamics model that combines processes of vanity and opinion propagation. The interactions take place between randomly chosen pairs. During an interaction, the agents propagate their opinions about themselves and about other people they know. Moreover, each individual is subject to vanity: if her interlocutor seems to value her highly, then she increases her opinion about this interlocutor. On the contrary she tends to decrease her opinion about those who seem to undervalue her. The combination of these dynamics with the hypothesis that the opinion propagation is more efficient when coming from highly valued individuals, leads to different patterns when varying the parameters. For instance, for some parameters the positive opinion links between individuals generate a small world network. In one of the patterns, absolute dominance of one agent alternates with a state of generalised distrust, where all agents have a very low opinion of all the others (including themselves)."

Each agent has a list of opinions about herself and other agents, ranging from -1 to +1. In this version, the parameter "condition" establishes the potential partners of each agent: global = all agents; local = agents within "influence_range"; and "small worlds" (see below).

NOTE: The article expresses the influence of j on i using the subscripts ij. Here, I use the subscript ij to express the influence of i on j.

ROUTINE:

STEP 1: Opinion Propagation.
Ask turtles, create an out-link "opinion" from turtle i to turtle j, and vice-versa, Oij and Oji. The "propagation coefficient" is called "intensity" here and is an attribute of opinions. The logistic equation is used. Intensity = 1 / (1 + exp (-1 * ((aij - aii) / sigma))), where aij is the evaluation of opinion i of j, and aii is the "self-regard" of agent i for herself. The basic idea expressed here is that "if i has a high opinion of j, then j is more influential." "Self-regard" is a turtle attribute. Sigma is a parameter which can be set in the interface.

The influence of j on i for i's self-regard is expressed as follows:
aii = aii + (intensityij * rho * (aji - aii + noise)), where aii is self-regard of agent i, and "rho" is a parameter 'ruling the importance of opinion propagation.' Now that the self-regard of agent i is updated, the "evaluation" of agent j by agent i (aij) is calculated as follows:
aij = aij + (intensityij * rho * (ajj - aij + noise)), where aij is the "evaluation" of opinion_ij, and ajj is the self-regard of agent j.

STEP 2: REPUTATION
A list is created for agent i of all of its out-opinion neighbors excluding itself and its current partner j. The "Evaluation" of opinion_iq is recorded, where q is the randomly selected neighbor. Then, the evaluation of opinion_iq is used to update or influence j's evaluation of q, opinion_jq, using the same equation above. The parameter "k_neighbors" determines the number of other agents about which agent i gossips- i.e. the # of reputations of other turtles i communicates to j.

STEP 3: VANITY DYNAMICS.
The idea is as follows: "agents tend to reward the agents that value them more positively than they value themselves and to punish the ones that value them more negatively than they value themselves." "Evaluation" of opinion ij is set as follows:
Eval_ij = Eval_ij + (w * (aji - aii + noise)), where "w" is the parameter "vanity." In words, agent i's new evaluation of j is the old evaluation plus the difference between j's evaluation of i and i's self-evaluation, plus some noise.

Recap:
Intensityij = 1 / (1 + exp (-1 * ((aij - aii) / sigma)))
aii = aii + (intensityij * rho * (aji - aii + noise))
aij = aij + (intensityij * rho * (ajj - aij + noise))
aij = aij + (w * (aji - aii + noise))

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